quadraticLimit_status
plain-language theorem explainer
In the small-strain regime the J-cost functional reduces to the standard kinetic action up to a bounded relative error of one tenth. This status declaration reports that the quadratic Taylor bound on the cost, the definition of kinetic action, and the derivation of Newton's second law are all in place with no remaining sorries or axioms. A physicist deriving classical mechanics from the Recognition Science cost functional would cite this report to confirm the small-strain bridge is established. The declaration is implemented as a literal string
Claim. The status of the quadratic limit section is that the pointwise bound $|J(1 + ε) - ε^2/2| ≤ ε^2/10$ for $|ε| ≤ 1/10$, the kinetic action $T[ε] = (1/2) ∫_a^b ε(t)^2 dt$, and Newton's second law $m γ̈ = -V'(γ)$ are all formalized with zero remaining proof obligations.
background
The module develops the quadratic limit of the J-cost in the small-strain regime where the strain γ equals 1 plus a small ε. The cost J(γ) = ½(γ + γ^{-1}) - 1 reduces to its quadratic Taylor expansion ½ε². This provides the bridge from the cost-functional formulation to standard kinetic-energy mechanics, with the J-action becoming the standard Lagrangian action and its Euler-Lagrange equation becoming Newton's second law, as described in the module documentation and the companion paper section on Newton's Second Law as a Corollary.
proof idea
This declaration is a scaffolding stub implemented as a string literal that reports the status of the module's main results. It enumerates the quadratic Taylor expansion theorem (a rebrand of the small-strain bound from the Cost module), the kinetic action definition, and the Newton's second law theorem, noting zero sorries and zero axioms.
why it matters
This status report marks the completion of the quadratic limit section, which shows how Newton's second law emerges as a corollary of the J-action in the small-strain limit. It supports the paper proposition in RS_Least_Action.tex on Newton's Second Law as a Corollary. In the Recognition Science framework this closes the bridge from the J-cost functional through the small-strain approximation to classical mechanics, consistent with the forcing chain that derives spatial dimensions and the action principle.
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